Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos,Isoparametric submanifolds and a Chevalley-type restriction theorem,Integrable systems, geometry, and topology,American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng,Ancient solutions to mean curvature flow for isoparametric submanifolds,Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres.
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Normalizing Flows on Tori and Spheres
Normalizing flows are a powerful tool for building expressive distributions in high dimensions. So far, most of the literature has concentrated on learning flows on Euclidean spaces. Some problems however, such as those involving angles, are defined on spaces with more complex geometries, such as tori or spheres. In this paper, we propose and compare expressive and numerically stable flows on such spaces. Our flows are built recursively on the dimension of the space, starting from flows on circles, closed intervals or spheres.
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- Award ID(s):
- 1841699
- NSF-PAR ID:
- 10216513
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 119
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 8083-8092
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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